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We develop a new method for visualizing and refining the invariances of learned representations. Specifically, we test for a general form of invariance, linearization, in which the action of a transformation is confined to a low-dimensional…
Geometric torsions are torsions of acyclic complexes of vector spaces which consist of differentials of geometric quantities assigned to the elements of a manifold triangulation. We use geometric torsions to construct invariants for a…
It is shown that the action for Hamiltonian equations of motion can be brought into invariant symplectic form. In other words, it can be formulated directly in terms of the symplectic structure $\omega$ without any need to choose some…
In its canonical formulation, general relativity is subject to gauge transformations that are equivalent to space-time coordinate changes of general covariance only when the gauge generators, given by the Hamiltonian and diffeomorphism…
There exist several ways of constructing general relativity from `first principles': Einstein's original derivation, Lovelock's results concerning the exceptional nature of the Einstein tensor from a mathematical perspective, and…
The structure of electrodynamics based on the variational principle together with causality and space-time homogeneity is analyzed. It is proved that in this case the 4-potential is defined uniquely. Therefore, the approach where Maxwell…
In this work we look for a geometric description of non-gravitational forces. The basic ideas are proposed studying the interaction between a punctual particle and an electromagnetic external field. For this purpose, we introduce the…
We discuss some equivalence relations between the non-relativistic quantum mechanics for particles subjected to potentials and for particles moving freely on background geometries. In particular, we illustrate how selected geometries can be…
Point particles fall freely along geodesics; strings do not. In string theory all probes of spacetime structure, including photons, are extended objects and therefore always subject to tidal forces. We illustrate how string theory modifies…
Using tangent bundle geometry we construct an equivalent reformulation of classical field theory on flat spacetimes which simultaneously encodes the perspectives of multiple observers. Its generalization to curved spacetimes realizes a new…
Variational principles in mechanics, field theory and geometric analysis are usually formulated on closed admissible classes, where boundary variations are either fixed or independently cancelled through natural boundary conditions.…
We consider the problem of whether there are deterministic theories describing the evolution of an individual physical system in terms of the definite trajectories of its constituent particles and which stay in the same relation to Quantum…
Quantum mechanics, one of the most successful theories in the history of science, was created to account for physical systems not describable by classical physics. Though it is consistent with all experiments conducted thus far, many of its…
We investigate the possibility that a background independent quantum theory of gravity is not a theory of quantum geometry. We provide a way for global spacetime symmetries to emerge from a background independent theory without geometry. In…
We consider the variational principle in the covariant formulation of modified teleparallel theories with second order field equations. We vary the action with respect to the spin connection and obtain a consistency condition relating the…
The link between 3D spaces with (in general, non-constant) curvature and quantum deformations is presented. It is shown how the non-standard deformation of a sl(2) Poisson coalgebra generates a family of integrable Hamiltonians that…
Systems of equations are invariant under "polydimensional transformations" which reshuffle the geometry such that what is a line or a plane is dependent upon the frame of reference. This leads us to propose an extension of Clifford calculus…
The aim of this paper is to show that the Lagrange-d'Alembert and its equivalent the Gauss and Appel principle are not the only way to deduce the equations of motion of the nonholonomic systems. Instead of them, here we consider the…
The field equations of a generalized $f(R)$ type gravity model, in which there is an arbitrary coupling between matter and geometry, are obtained. The equations of motion for test particles are derived from a variational principle in the…
It is widely accepted that the fundamental geometrical law of nature should follow from an action principle. The particular subset of transformations of a system's dynamical variables that maintain the form of the action principle comprises…